Subgame perfect equilibrium refines Nash equilibrium for sequential games. It ensures players act rationally at every decision point, considering future consequences. This concept is crucial for understanding strategic behavior in multi-stage interactions.

To find subgame perfect equilibrium, use backward induction. Start at the game's end and work backwards, determining optimal actions at each node. This process eliminates non-credible threats and yields strategies optimal for the entire game and every subgame.

Subgame Perfect Equilibrium

Properties of subgame perfect equilibrium

• Refinement of Nash equilibrium for sequential games ensures players' strategies are optimal for the entire game and every subgame
• Players act rationally at each decision point, considering the consequences of their actions (e.g., anticipating future moves)
• Assumes players have perfect information about the game structure and payoffs (e.g., knowing the game tree and possible outcomes)
• Strategies are sequentially rational, meaning they are optimal given the strategies of other players (e.g., choosing the best response at each stage)

Subgames in extensive form games

• A subgame is a portion of an extensive form game starting at a single decision node and including all subsequent nodes and branches
• To identify a subgame:
  1. Find a node where the player has perfect information about previous moves
  2. Include all nodes and branches following from that node
• Subgames must have a single starting point and contain all subsequent decision nodes and terminal nodes (e.g., a branch of the game tree)
• Proper subgames are subgames that do not include the entire game tree (e.g., a smaller portion of the game)

Equilibrium in sequential-move games

• To find the subgame perfect equilibrium, use backward induction:
  1. Start at the end of the game tree and work backwards
  2. At each decision node, determine the optimal action for the player, assuming all subsequent players will also act optimally
  3. Eliminate branches corresponding to suboptimal actions
• The resulting path from the root to a terminal node represents the subgame perfect equilibrium strategy profile
• In subgame perfect equilibrium, players' strategies are optimal at every decision point, considering the strategies of other players (e.g., making the best choice at each stage based on anticipated future moves)

Subgame perfect vs other equilibria

• Nash equilibrium (NE):
  • Applies to both simultaneous and sequential games
  • Strategies are optimal given other players' strategies, but not necessarily optimal for every subgame (e.g., may include non-credible threats)
• Weak perfect Bayesian equilibrium (WPBE):
  • For games with incomplete information (e.g., auctions with unknown valuations)
  • Requires players to update beliefs according to Bayes' rule and play optimally given those beliefs
• Sequential equilibrium:
  • Refinement of subgame perfect equilibrium for games with imperfect information (e.g., poker)
  • Requires players to have consistent beliefs about off-equilibrium path events and play optimally given those beliefs